The convolution of a discretely sampled signal can be expressed in matrix-algebra form as: EQU DV=B or DV-B=0
where V is the input signal to a system having transfer function matrix D, and, B is the system's output. (Matrices are indicated by a double top-arrow, vectors by a single arrow.) A commonplace problem in signal processing is, given a particular system output B, define the input V. The most straightforward solution to this problem is simply to calculate the inverse D.sup.-1 of transfer function D, and to calculate V directly. Unfortunately, if all the vectors of D are not linearly independent, or if D itself is only poorly known, this calculation is impossible. Moreover, system noise distorts signal V so that the apparent system input signal A is not readily deconvolved, i.e., DA-B=0 has no unique solution.
Tank and Hopfield have proposed a circuit (See D. W. Tank and J. J. Hopfield, "Simple Optimization Networks: An A/D Converter, Signal Decision Circuit, and a Linear Programming Circuit," IEEE Transactions on Circuits and Systems, Vol. Cas -33, No. 5, May, 1986) shown in FIG. 1, which is of interest to the deconvolution problem. The Tank-Hopfield circuit is an example of a neural network, i.e., a multiply connected array of amplifier circuits. The particular circuit of FIG. 1 was designed to minimize a cost function A.V (the dot product of vectors A and V) under the additional constraint that DV-B=0, i.e. that DV-B be minimized. A and V are nth order vectors, in which the components A.sub.i of A represent fixed costs, and the components V.sub.i of V represents variable costs; B is an mth order vector in which components B.sub.j of B represent constraints on the optimization of A.V, and D is an m by n order matrix whose elements D.sub.ji describe the informational interrelation between the cost function and the constraint function.
The Tank-Hopfield circuit of FIG. 1 contains two portions separated from one another by dotted lines and labeled, respectively, as the Variable Plane and the Constraint Plane. The inputs to the Variable Plane are the components A.sub.1 through A.sub.n of vector A, and the output the components V.sub.1 through V.sub.n of vector V. Similarly, the inputs to the Constraint Plane are the components B.sub.1 through B.sub.m of constraint vector B, and the outputs of the Constraint Plane have no significance other than as feedbacks to the Variable Plane. In FIG. 1, heavy dots indicate electrical connections, i.e. circuit nodes. Such nodes that are labeled with some D.sub.ji also have a conductance of magnitude D.sub.ji, through which flows each current incident upon the node. Each D.sub.ji represents the conductance among the inputs and outputs of the jth amplifier of the Constraint Plane and the ith amplifier of the Variable Plane, a negative sign associated with any D.sub.ji indicating negative feedback. The individual D.sub.ji 's represent the elements of matrix (or Kernel) D. Tank and Hopfield showed that the total energy dissipation (or power consumption), E, of the circuit shown in FIG. 1 is expressed by: ##EQU1## where D.sub.j is the jth column of matrix D, g is the transfer function of the Variable Plane amplifiers, R.sub.i is the ith resistor of resistors R.sub.1 through R.sub.n shown in RC network 30 of FIG. 1, and F is an error function indicating deviation from the condition DV-B=0. This equation is commonly referred to as the "energy function" or "Lyopunov function" of the system. The above expression for E assumes that the amplifiers' outputs dissipate energy only within the feedback loops of the circuit itself. This is a good assumption, because the outputs of the constraint plane amplifiers are not coupled to external, power dissipating components. The outputs of the variable plane amplifiers (i.e., V), need drive only circuitry that monitors voltage. Such circuits, like any well-designed voltage monitoring circuit, should be of very high input impedence so that the voltages monitored will be substantially unperturbed. The only requisites of F are that F(0)=0, and that F(z) for z.noteq.0 be a positive quantity. By inspection of FIG. 1, one can see that the circuit feeds back from the variable plane to the m inputs of the constraint plane, summing them there, with the inputs B. In this way the circuit power dissipation contains a term which depends on DV-B. Thus, the particular form of F is defined by the particular transfer functions of the constraint plane amplifiers. Tank and Hopfield showed that for a given D and B, the output V, will spontaneously adjust so that the energy function E tends towards (and reaches) a global minimum, minimizing cost function A.V under the given constraints. Especially noteworthy about the Tank-Hopfield circuit is that the constraint equation DV-B=0 under which the cost function is globally optimized is in the form of a convolution, suggesting that the Tank-Hopfield circuit can be modified to perform deconvolution of a discretely sampled signal without having to calculate the inverse D.sup.-1 of D.